Euler characteristics of general linear sections and polynomial Chern classes

TL;DR

This paper establishes a precise link between Chern-Schwartz-MacPherson classes and Euler characteristics of linear sections, extending formulas for polar degrees and polynomial Chern classes to broader contexts.

Contribution

It provides new formulas relating Chern classes and Euler characteristics, extends known results to arbitrary characteristic zero fields, and introduces a simple approach to polynomial Chern classes.

Findings

Derived formulas for degrees of polar maps of hypersurfaces.

Extended formulas to subschemes of higher codimension.

Proved polynomial Chern classes define homomorphisms to Z[t].

Abstract

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca-Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of characteristic~0, and proving a conjecture of Dolgachev on 'homaloidal' polynomials in the same context. We generalize these formulas to subschemes of higher codimension in projective space. We also describe a simple approach to a theory of `polynomial Chern classes' for varieties endowed with a morphism to projective space, recovering properties analogous to the Deligne-Grothendieck axioms from basic properties of the Euler characteristic. We prove that the polynomial Chern class defines homomorphisms from suitable relative Grothendieck rings of varieties to Z[t].